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August 16, 2012

Zeno's paradoxes

His (Xeno or Zeno for that matter) name actually came out over a short exchange of ideas between me and my friend over the internet regarding somebody or something that made the world, to some extent, ponder for a while. I was then moved and intrigued, that maybe I can also ponder a while on this not-so-ordinary person including his ideas especially on the things he was known for – his paradoxes.

Who actually is Xeno? The ever-reliable wikipedia.org further tells us that it is “Zeno” and not “Xeno”, although both are interchangeable because some guys would prefer the “X” for the “Z”. For purposes of discussion and to avoid confusion, let’s use what the wiki guys did – Zeno. As history has produced a handful of Zeno’s (from a bishop to a navigator), it is the Greek philosopher Zeno of Elea whom we are talking about. Zeno of Elea was born about 490 B.C. and continued his fruitful life until 430 B.C. As a philosopher, he was then, a bona fide member of the Eleatic School – school founded by another Greek pre-Socratic philosopher, Parmenides. A physical account of Zeno can be found through Plato’s work (Parmenides 127) describing our main man as “tall and fair to look upon.” Although there are details that were quite reliable, Greek biographers such Diogenes Laërtius, in his work (Lives of Eminent Philosophers) tells that Zeno was the adopted son of Parmenides and was likewise “skilled to argue both sides of any query presented.”

Ancient historians and philosopher believed that Zeno’s works were made, only to support that of Parmenides’. The Greek Neoplantonist philosopher Proclus described Zeno’s work contained arguments in which everything revealed nothing but contradictions. And since it was his predecessor Parmenides who started this way of reasoning, it was Zeno who developed the logical method called as “reductio ad absurdum”. Sounds absurd? Well, it really is because this logical argument means coming up with an absurd or an outcome which is completely ridiculous, for the sake of argument. It further goes on concluding that the premises on the argument are all wrong because of its absurd outcome. It is like proving something by contradicting it, in simpler terms. Looks and sounds funny, is it? Was Zeno behaving normal? Apparently, Zeno’s works and paradoxes have challenged, inspired, influenced and even enraged physicists, mathematicians and philosophers for over two millennia already.

But what’s in our Zeno that makes him known for his paradoxes? Let us first define what a paradox is. Dictionary.com defines paradox as a proposition or statement which is self-contradictory in nature, but expresses the truth in reality. (It really sounds, absurd eh?) A simple example of a paradox is the sentence “I am lying.” Assuming that my statement is true, and then it must be false. The statement is then false if it is agreed to be true. Got that one? And before we disagree with one another here, let’s sail on to Zeno’s world and explore it as far as our simple and uncontradicted minds can.

As was stated and discussed earlier, Zeno’s work mainly dealt on supporting his lover’s (Parmenides) ideas. (Just a side note – it was believed that it was during their time and age that affections and burst of feelings for the same sex emerged, thus giving birth to homosexuality. Example: the “Platonic Love”. And since there no concrete and profound documentations yet to be found, I’d rather deviate from it) Parmenides’ idea or philosophy states that reality is one and cannot be changed. At a glance, this is quite untrue since we can obviously see that change happen all around us every single time. Zeno further supported these notions by arguing that the concept of space and time cannot be continuous. In simpler terms, he argued that there was no such thing as movement. He furthered this argument by formulating the most famous paradox of his, the Tortoise and Achilles. A we all know, Achilles is that seemingly invulnerable hero of the Greek mythology.

Zeno goes on by stating that if Achilles were to challenge a tortoise to a race and the latter was given a head start, Achilles, with all his hero abilities (I hope you Warcraft fans are listening), cannot in any way overtake or outrun the good old tortoise. This is given on no matter how fast and how long he runs. Sounds absurd? Read on….

Although the above proposition may look and sound nonsense in reality; and that its for sure that our hero Achilles will eventually catch up with and overtake the tortoise, the impossibility or the paradox here lies in Zeno’s mathematical inspection of the race itself. Helium.com, helped us to translate in a more simple and understandable terms on Zeno’s brainchild. It further explained that in order for Achilles to travel to a certain distance (point A to point B), he has to move and cover the amount of distance between A and B. But in doing so, he must first cover at least half of its distance (point C, in this manner). And again, he is faced with the task of covering half of the distance to point C (point D, perhaps). But then again and again, his task gets so multiplied on trying to cover almost infinite number of points simply to get to point B! Without much thinking because you may already be dizzy by this time, it is evident that our poor Achilles can not move. Remember that the tortoise was given head start so the shelled mammal has already covered infinite number of points to which Achilles has to cover first (at least half of it!), but is hampered since his task is of covering half the distance is not over for he must cover at least “half of the half of the half of the half of the half…..” (The same idea is also Zenos’ basis in his second paradox which is the dichotomy paradox- by walking at a distance from a wall).

With the above paradox, the philosophical, scientific, mathematical and even the paranormal world launched an all out war to solve Zeno’s ideas. Many have or up to know, are trying to measure the distance between the two points. Some argue that, with the proper mathematical equations and computations, the tortoise cannot move as far as 2 meters! (ToeQuest.com). Quantum physicists would further tell that it was Zeno’s premise that was incorrect and expounded that it was not always necessary that in order to cover the distance of point A to point B, one first must travel at least half of its length, and so on and so forth. With that manner, Zeno was able to stop time and space for there were no movements or change that was evident.

Modern experts were not in anyway bothered by these ideas and in turn, tried to solve this ancient riddle. They offered the idea that these infinite numbers of points do have length, and are obviously measurable. Modern physicists argued that there are tiny groups of quantum mush and foams. These quantum specks are measurable once the movement of time and space are stopped from moving. So molecularly small as it can be that it somewhat challenged Max Planck to develop the Planck length (about 0.000 000 000 000 000 000 000 000 000 000 000 001 6 meters, in figures) in order to measure these small intervals.

The last of the three famous paradoxes of Zeno is the Arrow paradox. It is oftentimes called the Fletcher’s paradox. Zeno contradicts this event by premising that objects must change or must be able to move from its position in order to have motion. Philosophers would simply explain the paradox as that the arrow retains its own size and at rest whether or not it is in flight or at rest. It is like taking pictures of an arrow in flight. From a single frame, it can be gleamed that the arrow is not moving. It also is true on the other frames as well that the arrow is really at a state of rest. Therefore, there is no movement present. Helium.com further explains said paradox by adding that the arrow has, at any point during its flight, an exact location and therefore cannot move or cannot be in motion. This event makes sense when is viewed in the present and not in the past or in the future. And since the former explanation is given; it would be clear that, as Zeno puts it that the idea of Time consists of a series of present frames whereby each frame represents no movement or manifests a state of being stationary, thus the impossibility of movement is emphasized.

The arguments and discussions above about Zeno’s paradoxes influenced the way man perceived space and time. To Zeno, space, time and motion are impossible to exist. But with the modern studies and modern minds plunging into Zeno’s paradoxical pool, clear and factual possibilities emerges which enables the common modern man to have a grasp of his ideas and concepts.

For a simple reader or for a common man, all these may look ridiculous and may sound funny but worth learning and comprehending when closely scrutinized and examined upon. All it takes, as some would put it, is an open mind. It need not be absurd.